This worksheet is also part of one or more other Books. Modifications will be visible in all these Books. Do you want to modify the original worksheet or create your own copy for this Book instead?

This worksheet was created by '{$1}'. Do you want to modify the original worksheet or create your own copy instead?

This worksheet was created by '{$1}' and you lack the permission to edit it. Do you want to create your own copy instead and add it to the book?

Law of Sines

Start with [math]\Delta[/math]ABC and circumscribe it about a circle with center at [i]O[/i] and with radius equal to R units.[br]Draw in the diameter [i]CJ[/i] and the chord[i] BJ[/i]. <CBJ is a right angle, since it is inscribed in a semicircle.[br]Therefore in both figures,[br][math]sinJ=\frac{a}{CJ}=\frac{a}{2R}[/math] [br]In the first diagram <J=<A because they are both inscribed in the same arc of the circle.[br]In the second diagram <J=180-<A, because opposite angles of an inscribed quadrilateral are supplementary.[br]Remember that [math]sin\theta=sin\left(180-\theta\right)[/math] --> [math]sinJ=sinA[/math] in both figures. Therefore, [math]sinA=\frac{a}{2R}[/math]-->[math]\frac{a}{sinA}=2R[/math][br]The same procedure applied to the other angles of [math]\Delta[/math]ABC yields[br][math]\frac{b}{sinB}=2R[/math] and [math]\frac{c}{sinC}=2R[/math][br]Combining these results we get the extended Law of Sines,[br][i]For a triangle ABC with circumradius R,[br][math]\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=2R[/math][/i]